Correct Minkowski product; build chain of three spheres

engine-proto/engine.jl

@@ -78,7 +78,7 @@ end
 
 abstract type Relation{T} end
 
-mprod(v, w) = v[1]*w[2] + w[1]*v[2] - dot(v[3:end], w[3:end])
+mprod(v, w) = (v[1]*w[2] + w[1]*v[2]) / 2 - dot(v[3:end], w[3:end])
 
 # elements: point, sphere
 struct LiesOn{T} <: Relation{T}
@@ -94,7 +94,7 @@ struct AlignsWithBy{T} <: Relation{T}
   elements::Vector{Element{T}}
   cos_angle::T
   
-  LiesOn{T}(sph1::Point{T}, sph2::Sphere{T}, cos_angle::T) where T = new{T}([sph1, sph2], cos_angle)
+  AlignsWithBy{T}(sph1::Sphere{T}, sph2::Sphere{T}, cos_angle::T) where T = new{T}([sph1, sph2], cos_angle)
 end
 
 equation(rel::AlignsWithBy) = dot(rel.elements[1].vec, rel.elements[2].vec) - rel.cos_angle
@@ -166,15 +166,28 @@ end
 
 # ~~~ sandbox setup ~~~
 
-a = Engine.Point{Rational{Int64}}()
-s = Engine.Sphere{Rational{Int64}}()
-a_on_s = Engine.LiesOn{Rational{Int64}}(a, s)
-ctx = Engine.Construction{Rational{Int64}}(elements = Set([a]), relations= Set([a_on_s]))
+CoeffType = Rational{Int64}
+
+a = Engine.Point{CoeffType}()
+s = Engine.Sphere{CoeffType}()
+a_on_s = Engine.LiesOn{CoeffType}(a, s)
+ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
 eqns_a_s = Engine.realize(ctx)
 
-b = Engine.Point{Rational{Int64}}()
-b_on_s = Engine.LiesOn{Rational{Int64}}(b, s)
+b = Engine.Point{CoeffType}()
+b_on_s = Engine.LiesOn{CoeffType}(b, s)
 Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
-eqns_ab_s = Engine.realize(ctx)
+eqns_ab_s = Engine.realize(ctx)
+
+spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
+tangencies = [
+  Engine.AlignsWithBy{CoeffType}(
+    spheres[n],
+    spheres[mod1(n+1, length(spheres))],
+    -1//1
+  )
+  for n in 1:3
+]
+ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))